View source for Theory of computation

== Introduction ==

The '''theory of computation''' is a branch of [[computer science]] and [[mathematics]] that deals with how efficiently problems can be solved on a model of computation, using an algorithm. It encompasses the fundamental concepts of [[automata theory]], [[computability theory]], and [[complexity theory]]. The field seeks to understand the nature of computation and the inherent limitations of computational processes.

== Automata Theory ==

Automata theory is the study of abstract machines and the problems they can solve. It provides a formal framework for designing and analyzing computational systems. The primary models of computation studied in automata theory include [[finite automata]], [[pushdown automata]], and [[Turing machines]].

=== Finite Automata ===

Finite automata are the simplest type of automata and are used to model systems with a limited amount of memory. They are characterized by a finite number of states and transitions between those states. Finite automata are used in various applications, such as text processing, [[compiler design]], and [[network protocols]].

=== Pushdown Automata ===

Pushdown automata extend finite automata by adding a stack, which provides additional memory. This allows them to recognize [[context-free languages]], which are more complex than the regular languages recognized by finite automata. Pushdown automata are crucial in the design of [[parsers]] for programming languages.

=== Turing Machines ===

Turing machines are the most powerful model of computation and can simulate any algorithm. They consist of an infinite tape, a head that reads and writes symbols on the tape, and a set of states. Turing machines are used to define the concept of [[algorithmic computability]] and are central to the study of the [[Church-Turing thesis]].

== Computability Theory ==

Computability theory explores the limits of what can be computed. It addresses questions about which problems can be solved by algorithms and which cannot. Key concepts in computability theory include [[decidability]], [[reducibility]], and [[recursive functions]].

=== Decidability ===

A problem is said to be decidable if there exists an algorithm that can determine the solution to the problem for any input in a finite amount of time. Decidability is a fundamental concept in the theory of computation, as it delineates the boundary between problems that can and cannot be solved algorithmically.

=== Reducibility ===

Reducibility is a technique used to relate the complexity of different problems. A problem A is reducible to a problem B if a solution to B can be used to solve A. This concept is used to classify problems into complexity classes and to prove the undecidability of certain problems.

=== Recursive Functions ===

Recursive functions are a class of functions that can be defined by a set of base cases and recursive rules. They are used to formalize the notion of computation and are central to the study of computability. Recursive functions include [[primitive recursive functions]] and [[general recursive functions]], which differ in their expressive power.

== Complexity Theory ==

Complexity theory investigates the resources required to solve computational problems, such as time and space. It seeks to classify problems based on their inherent difficulty and to understand the trade-offs between different computational resources.

=== Time Complexity ===

Time complexity measures the amount of time an algorithm takes to solve a problem as a function of the size of the input. It is typically expressed using [[big O notation]], which provides an upper bound on the growth rate of the algorithm's running time. Common complexity classes include [[P]], [[NP]], and [[EXP]].

=== Space Complexity ===

Space complexity measures the amount of memory an algorithm uses as a function of the size of the input. Like time complexity, it is expressed using big O notation. Space complexity is an important consideration in the design of efficient algorithms, particularly for problems involving large data sets.

=== Complexity Classes ===

Complexity classes are sets of problems that share similar resource requirements. They provide a framework for understanding the relative difficulty of different problems. Key complexity classes include P, NP, [[co-NP]], and [[PSPACE]].

== Advanced Topics ==

The theory of computation also encompasses several advanced topics that extend beyond the basic models and concepts.

=== Quantum Computation ===

Quantum computation is a field that explores the use of [[quantum mechanics]] to perform computation. Quantum computers use [[quantum bits]] or qubits, which can represent multiple states simultaneously, potentially allowing them to solve certain problems more efficiently than classical computers.

=== Cryptographic Complexity ===

Cryptographic complexity studies the computational difficulty of problems related to [[cryptography]], such as [[encryption]] and [[decryption]]. It focuses on the hardness assumptions that underpin the security of cryptographic systems and the development of efficient algorithms for cryptographic tasks.

=== Interactive Proof Systems ===

Interactive proof systems are a class of computational models that involve interaction between a prover and a verifier. They are used to study the complexity of verifying solutions to problems and have applications in [[cryptographic protocols]] and [[zero-knowledge proofs]].

<div class='only_on_desktop image-preview'><div class='image-preview-loader'></div></div><div class='only_on_mobile image-preview'><div class='image-preview-loader'></div></div>

== See Also ==

* [[Formal Language]]
* [[Algorithm]]
* [[Computational Complexity Theory]]

== References ==

[[Category:Computer Science]]
[[Category:Mathematics]]
[[Category:Theoretical Computer Science]]